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Which of the following assertions is correct?

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CSIR UGC (NET) Mathematical Science: Held On (7 June 2023)
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  1. \(\rm\displaystyle\lim _n \sup e^{\cos \left(\frac{n \pi+(-1)^n2e}{2 n}\right)}>1\)
  2. \(\rm\displaystyle\lim _n ~e^{\log _e\left(\frac{n \pi^2+(-1)^n e^2}{7 n}\right)}\) does not exist.
  3. \(\rm\displaystyle\lim _n \inf ~e^{\sin \left(\frac{n \pi+(-1)^n 2 e}{2 n}\right)}<\pi\)
  4. \(\rm\displaystyle\lim _n ~e^{\tan \left(\frac{n \pi^2+(-1)^n e^2}{7 n}\right)}\) does not exist.

Answer (Detailed Solution Below)

Option 3 : \(\rm\displaystyle\lim _n \inf ~e^{\sin \left(\frac{n \pi+(-1)^n 2 e}{2 n}\right)}<\pi\)
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Detailed Solution

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Explanation:

 =  = 

So  =  = e0 = 1  1

Option (1) is false

Also  = = e < π

Option (3) is correct

Also  =  = 

So  =  which is finite

Option (2) is false

 =  which is finite

Option (4) is false

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